Minimum Phase Revisited Jeff Grossman, Gary Margrave, and Michael Lamoureux

Introduction Minimum phase for digital systems Fourier synthesis: an example Tempered distributions Causality and the Hilbert transform Minimum phase for analog systems Examples Conclusions

Minimum phase for Digital systems

Minimum phase for Digital systems A minimum phase digital signal… has all the poles and zeros of its Z-transform inside the unit circle of the complex plane is causal, stable, and always has a minimum phase convolutional inverse has its energy concentrated toward time 0 more than any other causal signal having the same magnitude spectrum

Minimum phase for Digital systems Every causal, stable, all-pole digital filter is minimum phase Fermat’s principle of least traveltime implies seismic wavelets are minimum phase Wiener and Gabor deconvolution assume that a constant-Q-attenuated seismic wavelet is minimum phase

Fourier synthesis an example

Fourier synthesis for a Random spike series

Fourier synthesized signals

Crosscorrelations with original signal

Fourier Synthesis So, it seems phase accuracy is more important than amplitude spectrum accuracy… But, in Gabor/Wiener deconvolution, a phase spectrum is computed from an estimate of the amplitude spectrum

Tempered distributions a class of generalized functions

Tempered distributions A way to make sense of divergent integrals A way to make sense of Dirac’s delta “function” A tempered distribution is a continuous linear mapping T : S(R)  C S(R) is the Schwartz space of smooth, rapidly decreasing functions, e.g., Gaussians

Tempered distributions, some examples Given a nice enough function f that doesn’t grow too quickly, defines a tempered distribution.

Tempered distributions, some examples Any function f for which defines a tempered distribution,

Tempered distributions, some examples Dirac’s delta function, is a tempered distribution that doesn’t correspond to any function.

Causality and the Hilbert transform

Causality and the Hilbert transform Given a tempered distribution, T, write it as a sum of its causal and anti-causal parts:

Causality and the Hilbert transform It turns out that the Hilbert transform is given simply as:

Minimum phase for Analog systems

Minimum phase for Analog systems

Examples

Examples

Why not?

Summary In the analog examples, was invertible, but was not “stable” (not integrable, or it had infinite energy). In each case, a convolution could reduce the original distribution to a spike, by division in the Fourier domain:

Summary Sometimes the Hilbert transform relation fails for analog minimum phase signals, but it works very well on bandlimited versions. We computed a simpler min phase spectrum than the original ringy one, yet reconstructed the minimum phase signal with very high fidelity.

Acknowledgements The CREWES Project and its sponsors The POTSI Project and MITACS My co-authors/supervisors My energetic son, for waking me up at night so I could think about wavelets.